见证 Kirkwood-Dirac 非正性的凸屋顶
Convex roofs witnessing Kirkwood-Dirac nonpositivity
AI总结：
本文为一般混合态构建了两个 KD 非正性见证：支撑不确定性的凸屋顶和总 KD 非正性的凸屋顶，后者作为忠实见证在纯态层面捕获了 KD 分布的非正性质。
摘要：
给定两个可观测量 $A$ 和 $B$，可以为每个量子态关联一个 Kirkwood-Dirac (KD) 准概率分布。KD 分布类似于联合经典概率，但它可以取负值或非实数值，这些值与态的非经典特征相关。在过去十年中，KD 分布作为一种多功能工具被推至前沿，用于研究和构建量子优势与非经典现象。KD 分布也被用于确定量子-经典边界。为此，必须拥有判断态何时为 KD 非正的见证。先前的工作已经建立了纯态相对于 $A$ 和 $B$ 的本征基的不确定性与 KD 正性之间的关系。如果这种 $\ extit{支撑不确定性}$ 很大，则该态不能是 KD 正的。在此，我们为一般混合态构建了两个 KD 非正性见证。我们的第一个见证是支撑不确定性的凸屋顶；它不是忠实的，但它将 KD 正性与小支撑不确定性之间的关系扩展到了纯 KD 正态的凸包上。我们的另一个见证是总 KD 非正性的凸屋顶，它为纯 KD 正态的凸包提供了一个忠实的见证。这意味着总非正性的凸屋顶在底层纯态层面上捕获了 KD 分布的非正性质。
英文摘要：
Given two observables $A$ and $B$, one can associate to every quantum state a Kirkwood-Dirac (KD) quasiprobability distribution. KD distributions are like joint classical probabilities except that they can have negative or nonreal values, which are associated to nonclassical features of the state. In the last decade, KD distributions have come to the forefront as a versatile tool to investigate and construct quantum advantages and nonclassical phenomena. KD distributions are also used to determine quantum-classical boundaries. To do so, one must have witnesses for when a state is KD nonpositive. Previous works have established a relation between the uncertainty of a pure state with respect to the eigenbases of $A$ and $B$ and KD positivity. If this $\textit{support uncertainty}$ is large, the state cannot be KD positive. Here, we construct two witnesses for KD nonpositivity for general mixed states. Our first witness is the convex roof of the support uncertainty; it is not faithful, but it extends to the convex hull of pure KD-positive states the relation between KD positivity and small support uncertainty. Our other witness is the convex roof of the total KD nonpositivity, which provides a faithful witness for the convex hull of the pure KD-positive states. This implies that the convex roof of the total nonpositivity captures the nonpositive nature of the KD distribution at the underlying pure state level.
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